Over the past 5 years or so, superconductivity has been observed in a number of different materials with an extremely low density of charge carriers. Some of these are even among the most dilute metals that exist, such as SrTiO3 and bismuth. In this limit, the screening of the Coulomb repulsion is poor, and moreover, the conventional phonon mechanism for superconductivity is completely irrelevant. This raises the question: how can these materials be superconducting? I will propose two different mechanisms for superconductivity in low-density metals, based on dynamically screened Coulomb interactions and fluctuations near a structural quantum critical point. I will further discuss the prospects of these mechanisms for topological superconductivity and superconductivity in two-dimensional materials.